I'm a little confused by the First Contact TV ep...are they supposed to contact the civilization right before they have their first warp flight, or are they supposed to have a big "Welcome to the Rest of the Galaxy" party after they go to warp?

Yes they travel back in time to visit that man that creates the first worp speed. I remember Gordi explaining to him about how famous he is going to be and he was talking about when he visited his statue when he was a school boy.